On the Martin Boundary of Weakly Coupled Balayage Spaces

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1 On the Martin Boundary of Weakly Coupled Balayage Spaces K. JANSSEN Mathematisches Institut, Heinrich-Heine-Universität Düsseldorf, Universitätsstr. 1 D Düsseldorf, Germany ( janssenk@uni-duesseldorf.de) Abstract. We characterize balayage spaces on countable sums by weak coupling of balayage spaces through a transfer kernel. We show that polyharmonic functions, completely excessive functions, and also branching balayage spaces can be studied in this setting. We classify extremal positive harmonic functions for balayage spaces on countable sums by extremal harmonic functions on subsums. Mathematics Subject Classification (2000): 31D05, 31B10, 31B30, 60J35, 60J45 Key words: Martin boundary, weak coupling, balayage space, polyharmonic function, completely excessive function, Hunt process on countable sums, branching property 0. Introduction In recent decades axiomatic potential theory developed as a powerfull tool to study properties of suitable elliptic and parabolic differential operators. It turned out, that many results carried over to the more general setting of pseudo-differential operators. The axiomatic setting covering these topics was the theory of balayage spaces initiated in [BH86] by J. Bliedtner and W. Hansen. Moreover, it was discovered, that also weakly coupled systems of balayage spaces can be studied in the framework of balayage spaces, hence, in particular, systems of weakly coupled (partial or pseudo-)differential operators enter into the setting of balayage spaces. In this paper we show that balayage spaces on a countable topological sum i E i (of locally compact spaces with a countable basis for the topology) are exactly the weakly coupled balayage spaces on the spaces E i. This is a simple extension of results obtained 1

2 by M. Meyer in [Me88] and corresponds to results obtained by Bouleau [Bou80] for right Markov processes on direct sums. We show that large classes of examples fit into the setting of weakly coupled balayage spaces: polyharmonic functions, completely excessive functions, and also branching balayage spaces associated with some nonlinear zero-order pertubations can be treated in this setting. Finally, we obtain some results concerning the Choquet-type integral representation of positive harmonic functions for weakly coupled balayage spaces. We show that extremal positive harmonic functions for weakly coupled balayage spaces are determined by extremal harmonic functions for subcouplings, and also a partial converse. By Choquet s integral representation this induces a decompostion of all positive harmonic functions. 1. Balayage spaces on direct sums We use freely the results from J. Bliedtner and W. Hansen in [BH86]. To fix notations, remember that a balayage space is determined by the following ingredients: U = (U λ ) λ 0 is a resolvent of proper kernels on a locally compact space E with a countable base of the topology and field E := B(E) of borel sets. pe denotes the set of positive numerical functions. S := S(U) := {s pe : λu λ s s for λ, s < U 0 -a.e.} denotes the class of positive superharmonic functions. (E, S) is a balayage space iff i) for every s S there exists an increasing sequence (s n ) S C(E) satisfying s n s, ii) there exist u, v > 0 in S with u v C 0(E), iii) E = σ(s) is the field generated by S. For every B E there exists a unique kernel H B on E such that for x E, s S we have H B s(x) = R Bc s (x) = inf{v(x) : v S, v s on B c }. Remember that h S is called harmonic on E iff h C(E) and H D h = h for every relatively compact open set D E. H := H(U) denotes the set of all harmonic functions on E. 2

3 A function p S is called a potential on E iff H Dn p 0 for an exhaustion of E by relatively compact sets D n. P := P(U) denotes the set of all potentials on E. Remember that 0 s S is called extremal iff s = u + v for u, v S implies u = αs for some α 0. S e denotes the set of all extremal elements of S. Every s S e is either an extremal harmonic function (i.e. s H e ) or an extremal potential (i.e. s P e ). In the sequel let I be a countable set with at least 2 elements. Let E = i I E i be a countable sum of locally compact spaces which have a countable base of the topology. We identify functions f on E with a column (f i ) i I of functions f i on E i for i I. A measure µ on E is given by a row (µ i ) i I of measures µ i on E i for i I. A kernel Q on E is given by a matrix (Q i,j ) i,j I of kernels Q i,j from E i to E j for i, j I, hence in the i-th row of this matrix we find for x E i the components of the measure Q(x, ) on E. Q is called a diagonal kernel if Q i,j = 0 for all i j. We shall characterize balayage spaces on topological sums. For the first result let (E, S(U)) be a balayage space on E = i I E i. Define the kernel Q on E by Q(x, ) := H Ei (x, ) for x E i, i I, and denote V := U QU. Obviously, for x E i the measure Q(x, ) does not charge E i for i I. Theorem 1.1 V is a diagonal kernel. V is the potential kernel of a resolvent V on E which is subordinated to U (i.e. V λ U λ for all λ 0). (E, S(V)) is the balayage space given by (E i, S(V i,i )). i I Moreover, U 0 = n 0 Qn V 0, and for some strictly positive f 0 C(E) we have Qf P(V) C(E) for all f pe with f f 0. 3

4 A converse is obtained in the following way: Theorem 1.2 Let V = i I Vi,i be a diagonal resolvent on E = i I E i, such that (E, S(V)) is a balayage space (equivalently, for every i I we have a balayage space (E i, S(V i,i ))). Moreover, let Q be a transfer kernel on E for V, i.e. for i I, x E i the measure Q(x, ) does not charge E i, and for some strictly positive f 0 C(E) we have Qf P(V) C(E) for all f pe with f f 0. If U := n 0 Qn V is a proper kernel then U is the potential kernel of a resolvent U on E such that (E, S(U)) is a balayage space. Proof of (1.1) and (1.2) For the special case of I = 2 and submarkovian resolvents U and V the result is just theorem (4.4) from [Me88] by M. Meyer. The assumptions concerning submarkovian can be ommited by using a Doob transformation and the fact, that for a potential kernel Q with Qf 0 P(V) C(E) we have obviously Qf specifically below Qf 0 for all 0 f f 0. The proof of (1.1) is obvious for general countable I, since measures and kernels on E decompose obviously. To prove (1.2) for general countable I, one can first use proposition (4.1) from [Me88] by M. Meyer in the case that Q is a triangular kernel; for general Q one can in an intermediate step for a suitable triangular kernel Q construct a balayage space on a much bigger direct sum, and then obtain the wanted result by collapsing the bigger direct sum exactly as it was done in the proof of theorem (4.4) from [Me88] by M. Meyer. The fact, that a proper kernel U := n 0 Qn V is the initial kernel of a resolvent, is also shown in proposition of [BeB04] for proper submarkovian resolvents V. Remark 1.3 a) Assume 1 S(U) in (4.1) and (4.2). Then there is a nice Hunt process X associated with U. Moreover, killing X whenever it leaves one of the spaces E i, we obtain a Hunt process Y associated with V. Using the results from [J06] it is easily seen that the transfer kernel Q decomposes as Q = W B, where B is a submarkovian kernel such that for i I and x E i the measure 4

5 B(x, ) does not charge E i, and W is a potential kernel for V given by W f(x) = E x (f Y T ) for the lifetime T of Y. Moreover, W is associated with the regular potential p given by p(x) := P x (Y T E) (x E), and the jump kernel B satisfies Bf(x) = E x [f X T X T = x] for x E and f pe. b) If the diagonal operator L := V 1 denotes the generator of V and if Q = V B, then the generator L of U is given by L = L + B; more explicitely, this reads for u = (u i ) i I as Lu = (L i u i + j B i,j u j ) i I (c.f. the paper by Boboc and Bucur [BB81] for details). 2. Examples (2.1) Polyharmonic spaces Let E := n i=1 (E 0 {i}), let V be a diagonal resolvent such that each V i,i is the resolvent associated with a harmonic space. Let Q := V B, where B is given by B((x, i), ) := ε (x,i+1) for i < n and B((x, n), ) = 0 for x E 0. If U := n 1 k=0 (V B)k V is a proper kernel, then U is the initial kernel of a resolvent of a balayage space. More general polyharmonic spaces appear if Q = W B for some continuous potential kernel W for V (c.f. detailed results by A. Boukricha in [Bou84]). Much earlier, polyharmonic functions were studied in classical potential theory by M. Niculescu in [Ni36], and in an axiomatic setting of Brelot harmonic spaces by E.P. Smyrnelis in [S75]. It was N. Bouleau, who recognized, that the notions of poly-potential theory reduce to usual potential theoretical notions on an enlarged state space (c.f. [Bou80]). 5

6 (2.2) Completely excessive functions Let V 0 be the resolvent of a balayage space on E 0, let V be the diagonal resolvent on E := i N (E 0 {i}) such that V i,i = V 0 for all i N. Assume that (V 0 0 )n is a proper kernel for every n N. Define the transfer kernel Q = V 0 B for B((x, i), ) := ε (x,i+1) for x E 0, i N. Then it is obvious from (1.2) that U 0 space. := n 0 Qn V 0 is the potential kernel of a resolvent U associated with a balayage From the construction of U it is easily seen, that a function h on E is harmonic if and only if h E0 is completely excessive with respect to the resolvent V 0, c.f. the papers by L. Beznea [Be88] and by H. Ben Saad and K. Janssen [BSJ90]. There it was shown (in the more general setting of basic resolvents), that extremal completely excessive functions are just the eigenfunctions. In the classical setting of E 0 = R + and V 0 0 (x, ) = 1 [x, [ λ, completely excessive is just the classical notion of completely monotone. (2.3) Branching balayage spaces Let V 0 be the resolvent of a balayage space on E 0. We assume V 0 0 = 0 P t dt for a strong Feller semigroup, i.e. P t f C(E 0 ) for every f pe below some strictly positive u S. Then it is known from corollary (5.4) in [BH86] that the i-fold product semigroup P t P t... P t is the semigroup of a balayage space on E i := i k=1 E 0. Denote by V i,i the associated resolvent. We shall construct a branching resolvent on E := i N E i. Therefore let B be a kernel from E 0 to i 2 E i, i.e. we have B = i 2 B i, where each B i is a kernel from E 0 to E i. It is well known that B extends uniquely to a branching kernel B on E, i.e. B has the following branching property: define for f pb(e 0 ) the function f pb(e) by f(x 1,..., x i ) := i k=1 f(x k) for x = (x 1,..., x i ) E i ; then B f = Bf. If V denotes the diagonal resolvent on E associated with V i,i, then theorem (1.2) yields a balayage space (E, S(U)) for the transfer kernel Q := V ˆB, provided Q is a proper kernel. The standard example is given by B(x, ) = i 2 q i (x) i k=1 ε x, for (q i ) pe 0 satisfying i 2 q i 1 (c.f. below for the probabilistic interpretation). 6

7 If L denotes the generator of V 0 on E 0 then the generator L of U satisfies in this case ( L f) E1 = Lf + i 2 q i f i for f 1 in the domain of L. Consequently, in this way some nonlinear 0-order pertubations of L can be studied (c.f. M. Nagasawa s paper [Na76] for more details and more references). A variant of this construction obtains, if we divide each E i by the group of permuations of coordinates. Then E can be identified with the set Ẽ := {µ : µ is a σ-finite measure on E such that µ(a) N 0 for all A B(E)}. Details of this construction were studied in the setting of Hunt processes by Nagasawa in [Na76], where the following probabilistic interpretation of the associated branching process is given: a particle starts moving in E 0 according to the process associated with V 0. At the lifetime T of this process the particle splits into a random number n of particles at random places in E 0, and then every particle moves on according to the n independent processes associated with V 0 and so on. Continuous branching was studied by Fitzsimmons in [F88]. The associated state space there is the set of measures on E, which is i.g. not locally compact, hence this does i.g. not fit into the setting of balayage spaces. In Dynkins book [D02] continuous branching was applied to study nonlinear partial differential equations. 3. Martin boundary Let S := S(U) be a balayage space on the countable topological sum E = i I E i. Denote by S(V) = i I S(Vi,i ) = i I Si for S i := S(V i,i ) the associated balayage space and let Q be the associated transfer kernel from 2. Moreover, let H e := H e (S) and He i := H e (S i ) for i I. For s S we denote τ s := {B E : H B s s}. For basic resolvents it is well known, that τ s is a filter if and only if s S e. For s S e the filter τ s is called the filter of cofine neighborhoods of s (this name is motivated by simple parabolic examples, where τ s is obviously related with the fine topology of the adjoint 7

8 harmonic space). τ s has a nice probabilistic interpretation: The Hunt process X s associated with the Doob-transformation by s converges along the filter τ s at it s lifetime. It is easily seen, that for h H e we have K c τ h for every compact subset K of E. Consequently, H e can be interpretated as a boundary of E, denoted as the Martin boundary E associated with U. Proposition 3.1 Let h H e satisfy Qh h. Then there is a unique i I such that E i τ h. Moreover, h 0 := h Qh H e (V), h 0 Ei H e (V i,i ), and h = k 0 Qk h 0. Proof. If Qh(x) h(x) for some x E i for some i I, then H Ei h(x) = Qh(x) h(x), hence E i τ h. Since τ h is a filter, i is uniquely determined. Since Q is the transfer kernel to produce the subordinated resolvent, we conclude h = h 0 + Qh for h 0 S(V), and this implies h 0 H(V). From U 0 = k 0 Qk V 0 we deduce h = k 0 Qk h 0. To prove h 0 H e (V) let u 1, u 2 H(V) such that h 0 = u 1 + u 2. Then h i := k 0 Qk u i S(U) for i = 1, 2. Consequently h = k 0 Qk h 0 = h 1 +h 2, hence h 1, h 2 are proportional with h, hence u i is proportional with h 0 for i = 1, 2 and h 0 H e (V). Proposition 3.2 Conversely, let h 0 H e (V), or equivalently, h 0 Ei H e (V i,i ) for some i I. If h := k 0 Qk h 0 < then h H e (U) and E i τ h. Proof. Let h 0 H(V). If h := k 0 Qk h 0 is finite, then h S(U) due to the fact that U 0 = n 0 Qn V 0. We conclude h H(U), since otherwise h 0 = h Qh could not be in H(V). If h 0 H e (V) then h = h 1 + h 2 for h 1, h 2 H(U) implies h 0 = u 1 + u 2 for u i := h i Qh i and u i H(V) for i = 1, 2. From h 0 H e (V) we conclude that u 1 and u 2 are proportional with h 0, hence h 1 and h 2 are proportional with h, i.e. h H e (U). 8

9 Remarks 3.3 a) Similar statements obtain, if we decompose E in a different way: Let I = J K be a decomposition of I. Then E = F G for F := i J E i, G := i K E i, and the above applies to U and the associated diagonal resolvent ( ) WF 0. 0 W G If h H e, then there are two possibilities: i) For some finite J I we have i J E i τ h. Then there is some minimal such J I and h is determined by some h 0 H e (W F ) for F := i J E i similarily to (3.1). Conversely, every h 0 H e (W F ) determines some h H e provided h 0 is sufficiently finite as in (3.2). Denote by He J the set of all extremal U-harmonic functions, for which J is the minimal finite subset of I such that i J E i τ h. ii) If we have H D h = h for D := i J E i for every finite J I, then h can not be related with harmonic functions of substructures as in i). Denote by He the set of all these extremal U-harmonic functions. b) Due to the general theory concerning Choquet type integral representations of superharmonic functions (c.f.[bbc81]) we conclude that every h H decomposes uniquely as h = h J + h, J I, 0< J < where h and each h J are (up to normalization) uniquely determined mixtures of elements of He and He J, respectively. Application 3.4 Let n = I < and assume that the transfer kernel Q is triangular. Then we have: a) For h H(U) there exists a h 0 H e (V) such that h = n 1 k=0 Qk h 0. b) For h 0 H e (V) such that h := n 1 k=0 Qk h 0 is finite we have h H e (U). Consequently, the Martin boundary E of E with respect to U is a subset of i I E i. Proof. Since Q is a triangular kernel, we have Q n = 0, hence every function f 0 satisfies Qf f. Then the result follows from (3.1) and (3.2). 9

10 Consequence 3.5 Due to the general theory concerning Choquet type integral representations of superharmonic functions (c.f. [BBC81]) we conclude that under the assumptions of (3.4) every harmonic function h H(U) has a representation as n h(x) = P u(x) dµ i (u) E i i=1 where µ i is a unique measure on the set H 1 e(v i,i ) of suitably normalized extremal harmonic functions for E i, and P u := n 1 k=0 Qk u. In the following very simple setting some of the phenomena described in (3.1), (3.2), and (3.3) show up. Example 3.6 Let a R { }, let E := E 1 E 2 := ( ) ( ) ], a[ {1} ], a[ {2}, and consider the diagonal resolvent V on E determined by uniform motion to the right on E i, i.e. V i,i f i (x, i) = a x f i (y, i) dy (i = 1, 2). Extremal harmonic functions for V are the multiples of g := 1 E1 and h := 1 E2. We consider the following simple coupling of V 1,1 and V 2,2 : Denote by B the kernel on E given by B((x, i), ) := ε (x,3 i) for x ], a[ and i = 1, 2, and let the transfer kernel Q be given by Q := V B. Then U := n 0 Qn V is the potential kernel of a resolvent U associated with a balayage space (E, S) according to (1.2). Harmonic functions w.r. to U are solutions u = ( u 1 u 2 ) of the system First case: a <. u 1 + u 2 = 0 u 2 + u 1 = 0 In this case (3.2) gives by simple computation two extremal harmonic functions w.r. to U, namely u and v given by u := n 0 Q n g = { sinh(a x) for (x, 1) E1 cosh(a x) for (x, 2) E 2 v := { cosh(a x) for (x, 1) Q n E1 h = n 0 sinh(a x) for (x, 2) E 2 10

11 Moreover, E 1 τ u and E 2 τ v and u Qu = g, v Qv = h holds. In particular, (3.1) applies and reproduces g and h from u and v. Second case: a =. In this case we have Qg = = Qh, and there exists (up to a factor) a unique extremal harmonic function w w.r. to U, namely w(x, i) = e x for x R and i = 1, 2. We have Qw = w, hence E i τ w for i = 1, 2. The following observation may appear as a curious fact: For x R we have sinh(a x) lim = e x = lim a sinh(a) a cosh(a x), cosh(a) i.e. after suitable normalization both extremal harmonic functions for a < converge to w with a. Similar examples on 3 E := ], a i [ {i} with a 1 = a 2 = and a 3 < show, how (3.3) helps to decompose H e (U). i=1 References [BB81] N. Boboc, G. Bucur: Pertubations in excessive structures. Springer Lecture Notes in Math (1981) p [BBC81] N. Boboc, G. Bucur, A. Cornea: H-Cones. Springer Lecture Notes in Math. 494 (1975) Order and convexity in Potential Theory: [Be88] L. Beznea: Ultrapotentials and positive eigenfunctions for an absolutely continuous resolvent of kernels. Nagoya Math. J. 112 (1988) p [BeB04] L. Beznea, N. Boboc: Potential theory and right processes. Mathematics and its Applications, 572. Kluwer Academic Publishers, Dordrecht (2004) 11

12 [BH86] J. Bliedtner, W. Hansen: Approach to Balayage. Springer Verlag (1986) Potential Theory. An Analytic and Probabilistic [BM71] N. Boboc, P. Mustata: Considérations axiomatiques sur les fonctions polysurharmoniques. Rev. Roum. Math. Pures et Appl. 16 (1971) p [Bou80] N. Bouleau: Espaces biharmoniques et couplage de processus de Markov. J. Math. Pures et Appl. 59 (1980) p [Bou84] A. Boukricha: Espaces biharmoniques. Springer Lecture Notes in Math (1984) p [BSJ90] H. Ben Saad, K. Janssen: Bernsteins theorem for completely excessive measures. Nagoya Math. J. 119 (1990) p [D02] E.B. Dynkin: Diffusions, superdiffusions and partial differential equations. American Mathematical Society Colloquium Publications, 50 (2002) [EK02] M. El Kadiri: Frontière de Martin Biharmonique et Représentation Integrale des Fonctions Biharmoniques. Positivity 6 (2002) p [F88] P. Fitzsimmons: Construction and regularity of measure valued Markov branching processes. Israel J. Math 64 (1988) p (corrections: 73 (1991) p. 127) [H03] W. Hansen: Modifications of balayage spaces by transitions with application to coupling of PDE s. Nagoyo Math. J. 169 (2003) p [I70] M. Itô: Sur les fonctions polyharmoniques et le problème de Riquier, Nagoya Math. J. 37 (1970) p

13 [J06] K. Janssen: Factorization of excessive kernels. New Trends in Potential Theory, Theta (2005) p [Me88] M. Meyer: Balayage Spaces on Topological sums. Potential Theory (1988) Proceedings of a conference on Potential Theory 1987 in Prague p [Mü88] H.-H. Müller: Subordination for Balayage spaces. Potential Theory (1988) Proceedings of a conference on Potential Theory 1987 in Prague p [Na76] M. Nagasawa: A probabilistic approach to nonlinear Dirichletproblem. Springer Lecture Notes in Math. 511 (1976) p [Ni36] M. Niculescu: Les fonctions polyharmoniques. Hermann 1936 [S75] E.P. Smyrnelis: Axiomatique des fonctions biharmoniques. Ann. Inst. Fourier 25 (1975) p (Ann. Inst. Fourier 26 (1976) p. 1-47) 13